The approximation theory has its origin with the concept of injective envelopes and
it has had a wide development in the context of module categories.
Auslander and Buchweitz [1]} studied the ideas of injective envelopes and
projective covers in terms of maximal Cohen-Macaulay approximations for certain
modules. Based on Auslander-Buchweitz's idea, Hashimoto [2]} defined the so called “Auslander-Buchweitz context" for abelian categories, and provided a new framework to homological approximation theory.
In [3]}, they introduced and developed an analogous of the Auslander-Buchweitz
approximation theory in the context of triangulated categories by using a version
of relative homology.
